APPENDIX A

No measurement is meaningful unless accompanied by an estimate of its uncertainty or error. Also no measurement is exact. There is always an error associated with an experimental measurement. That is not to say that scientists make errors or are uncertain about how to make their measurements. Instead the experimental equipment and measuring instruments have limitations that prevent us from determining a measurement with sufficient accuracy that we get the same exact number each time the measurement is repeated. The cm rule only has markings to the nearest mm, and the Spring Scale is marked to the nearest 0.5 Newton. When you read these scales you must estimate the last significant figure. Repeated measurements such as these which do not quite give the same result have random errors associated with them. If you figure the average of 10 measurements which you have repeated, then the 10 measurements will be randomly distributed both higher and lower in value about the average. The range of the deviation of the 10 measurements from the average is a measure of the random error associated with a single measurement. 

Of course, a scientist does occasionally make a mistake or a blunder, as human error can occur in reading scales or recording data. But such blunders are not considered "errors", and should be avoided in making your measurements and recording your data. 

An error coveys the sense of reproducibility of your measurements, as well as their accuracy. You will not be able to evaluate the comparison between your measurements and your model unless you know the range over which the measurements apply. There are several ways you can figure your random errors. Estimating the errors in your measurements is frequently the most difficult, yet most important, part of a scientific experiment. 

Aside from a mere "guesstimate" for the measurement error which you may sometimes need to make, the percent error is the simplest kind to figure and to quote. For most experiments you perform here, your percent errors will be in the range of a few to 50 percent. That is to say that a single measurement differs from either a mean value or a previously determined value (sometimes called "the accepted value") by the percent error, which is given by: % error = (avg - meas value)/avg. 

Although the percent error is occasionally given, normally the random error quoted is the "standard" or "probable" error. Examine your Lab Book where you listed individual measurements for the scale readings for the fishing weights. You may have taken five readings for one of the 6 oz fishing weights and listed them in a table.

For the measurements in the table above the expression for the standard error of measurement is given as

where the numerator under the square root sign is the sum of the squares of the numbers in column three of the table, (Pull-Average). The average is the sum of the measurements in column two divided by the number of measurements, five, and is found from column two in the table above to be 1.63 ± 0.02 . Once the average is known you can then find the numbers to enter in the third column. The standard error represents roughly 2/3 of the total range of the individual measurements from the average value. This fraction of the full range is based on statistics theory, and most scientists give the standard error when presenting their measurements. You should also always quote the standard error as your random error when you have sufficient individual measurements. Whatever error you list with your measurements, you should always say what kind the error is that you are quoting, namely, a percent error or a standard error. What happens to the size of the standard error in your average value if you increase the number of measurements you make by a factor of 10? a factor of 100? a factor of 10,000?

The standard error we figured above is associated with the average of a set of N individual measurements. What if we merely wanted to find the difference between the pulls on the Spring Scale exerted by two different fishing weights. We would measure one weight to pull with 1.61 Newtons and the other weight with 1.63 Newtons. How would we know whether this difference is real? We would need to combine the errors of the averages of the two measurements in some way.

The way to combine errors when adding (or subtracting) measurements is to sum the squares of the individual errors and take the square root. That is, if the errors were 0.002 and 0.003 for the two fishing weights, and we want to find the error in the difference between the pulls the two weights exert on the Spring Scale, then the error in this difference is given by,

Note that the error in the difference of two measurements is always larger than the error in an individual measurement. Also the error is the same whether you add the measurements or subtract them. In this example, what error do you find for the difference between the pulls of the two fishing weights? Is this difference significant? Why or why not?

You are not only interested in knowing how accurate your individual scale readings are, but you would really like to know how accurate the value of the slope of the line you figured is. The slope of the line, k, is a fundamental characteristic of the spring that you measured. If you were to measure the "spring constant" for another spring, how would you be able to tell whether the two springs have the same characteristics or not? Without knowing the precision with which you determined k, you would not be able to decide because the measuring errors in the scale readings on both the Spring Scale and the spring extension propagate through you analysis to the slope of the line you drew through the data points. How, then, can you combine the errors in the two sets of measurements to figure the effect of these errors in the slope, k?

The general rule, again from the theory of statistics, is that when you multiply or divide two measurements (or their averages), their errors combine as the square root of the sums of all of the errors (ratioed with their averaged values) associated with the measurements. We can write the ratio of the combined errors to the measured value of k symbolically as,

This formula will give you one value for the ratio ek/k. Since you measured k, you can then figure what ek is. Note that combining two measurements gives a combined error in the slope k, ek which is always larger than the error in either measurement. How would you find the combined errors of the sum of three measurements?

Always remember that errors are merely estimates of the accuracy of your measurements, and that the errors discussed above refer to the random, unavoidable errors inherent in all experimental measurements. Blunders or human error referenced earlier should have been removed from your measurements, if you recognize in advance that you made a mistake. If blunders enter your table of measurements without your awareness (for example, making a scale reading of the Spring Scale with your eye level far above ( or below) the horizontal, so that "parallax error" is introduced into your measurements, you may never be aware of your mistake. In such cases the data point may appear discrepant compared with the other values.

Never discard a discrepant point, however, without knowing that it has arisen from a blunder of some kind. Perhaps the one discrepant point is the correct one, and you introduced parallax in the other four measurements. It is of course highly unlikely that you make so many blunders in your measurements. But the message should be clear. Never discard measurements without a very good reason.

In addition to random errors, there are "systematic" errors which can enter your data, and they sometimes cause immense problems, and the source or sources of systematic errors can be very difficult to identify. One source of a systematic error which could enter your scale readings for the Spring Scale could be misalignment of the marker from the zero point when there is no load on the spring. To check against this systematic error, you were supposed to have made scale readings both before and after you weighed the fishing weights. The way to handle systematic errors is to avoid them at all costs. When designing your experiments, you should anticipate all possible sources of systematic errors that you can think of, and plan the experiment to avoid or at least minimize their effects.

Remember that a measurement without an error estimate is meaningless. You will find that you will be unable to discuss your experimental results without knowing the precision of your measurements.
 

Handling Graphs

Usually the most convenient way to examine your experimental results is not by inspecting your Data Table which summarizes your measurements. Rather, as we have seen, by plotting the data values of two variables in a graph. We have already seen that the graph of pull versus extension led us to conclude that the strength of the pull of a fishing weight on the spring was directly proportional to the amount that the spring lengthened due to the load. The constant of proportionality you derived was the slope of the graph, k. 

Your graphed results may not always show a correlation between two variables that give a straight line, or any line at all. Sometimes you may find that the two variables you plot, in fact, show no obvious trend or pattern at all. In fact there are occasions when you plot two variables measured in an experiment and hope that no pattern is evident in the graph, because the two variables should be totally unrelated to each other. Suppose, for example, that you wanted to make sure that the Spring Scale readings of one of the 6 oz fishing weights did not depend on the time of day that you read the scale. You were afraid that you might have inadvertently introduced a systematic error in your readings of the spring scale because you were not as awake early in the morning when you began your readings, as later in the morning after two cups of coffee. 

Can you design an experiment where you would test this hypothesis? What variables would you measuring? Which is the independent and which the dependent variable? This experiment was carried out by a student who made scale readings at various times during one recent morning. What would you conclude from her data points which are graphed below?

How accurate was her clock? With what accuracy do you think she could read the spring scale? Was her hypothesis correct? Why or why not?

This graph is sometimes called a "scatter graph" because it shows no correlation between the time of day and the amount of pull exerted by the fishing weight. Sometimes it is useful to plot two variables in such a graph to demonstrate that there is no correlation between the two variables. This technique is frequently used when searching for systematic errors in experimental data. What characteristic of the graph tells us this? When searching for systematic errors in your data, you are always happy not to find a correlation evident in your graphs.

Occasionally a plot of your data points may show a graph like the one below. How would you interpret the relation between the two variables? Here we have measured the spring diameter after loading on

various fishing weights. What would you conclude from this graph? Is there a correlation between the Spring Diameter and the Pull Strength of the spring? If so, can the correlation best be modeled with a straight line or curved line?

Not all correlations found between variables in experiments result in straight lines. Straight line correlations are the simplest to interpret. But occasionally the data points indicate a non-linear correlation.

Conversion of Curved Functions to Straight Lines

Frequently in science the experimental data when plotted cannot be modeled with a straight line. Instead a curved function such as an exponential or a parabolic (quadratic) function is a model which conforms to the experimental data better. In fact if you were to model the experimental data, positions (or distance) vs. time, which you determined for the falling tennis ball above, with a variety of curved functions, you would find that the curve that provided the "best fit" to your data is a parabolic function with the form

This equation is the model which provides the "best-fit" to the d vs. t curve for any falling object on this planet, as long as air resistance has a negligible effect on the object’s downward motion. The positions, d, represent the dependent variable, and time the independent variable in this experiment, whereas a is a constant which represents the acceleration due to the gravitational force at the surface of the Earth. This acceleration is approximately constant at any place on the surface of the Earth, and is usually given the symbol g. Since the shape of the Earth is not quite spherical and there are both mountains and valleys, this gravitational constant varies slightly over different parts of the planet’s surface. But within the experimental errors here, g is the acceleration of an object falling toward the Earth due to the Earth’s force of gravity. Do the data in your graph conform within the experimental values with an expression of this form? If so, how should you proceed to find the value of g from your data?

The reason we would like to know the value of g is that the expression above can then, be used to predict the position of an object at any time as it falls toward the ground. Note that the expression  is a simple, empirically determined model for a falling ball. The model does not contain nor use any theory about what gravity is or how it causes the motion. We do not need to know anything about the gravitational force in order to predict the position of a falling object as a function of time. This is a very powerful aspect of science. We do not need to understand the physical phenomenon to come up with a scientific law or model! We only need to be able to predict the phenomenon.

How then can we determine the gravitational constant of acceleration, g, for the Earth from a data set consisting of positions, d, and times, t ? The expression for a parabola,  is then our model for the falling ball phenomenon. We need to know the value of the constant g in order to evaluate the expressions for any time. In other words we need to know g in order to use our model to prediction the position of the ball at any time. The graph of d vs. t is a parabola which passes through the origin of the graph, at d=0 and t = 0.

A convenient trick frequently used in science when exponential or parabolic functions are used to model data is to plot the logarithms of variables against each other. What is the significance of the slope of the graph when the logs are plotted?

If we take the log to the base 10 of both sides of the expression for our model for the falling ball, we find

If we separate the constant terms from the variable terms, we find

This expression has the form

where c is a constant (= ) that corresponds in a graph to the y intercept or the value of y when x=0, and the slope of the line in the above expression is 2. Therefore by plotting the logarithmic values of the data points, log d vs. log t, and fitting a line with a slope of 2 through the plot, the value of g can be found from the y intercept of the graph. Note that we cannot use the first set of data points in the data table, since the log of zero is undefined. We simply omit that first d,t pair, and proceed by plotting the remaining log d and log t values from our Data Table in the graph of log d vs. log t.

    

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